Math 311-102 (Summer 2006)
Name
1
Test II
Instructions: Show all work in your bluebook. Cell phones, laptops, calculators that do linear algebra or calculus, and other such devices are not
allowed.
1. (10 pts.) Determine whether or not S = {f ∈ C[2, 5] | f (2) = f (5)}
is a subspace of C[2, 5].
2. (5 pts.) For the subsets of P3 below, which cannot be linearly independent? Which cannot span? Briefly explain how you are getting
your answers.
(a) {1 − 2x, x2 , x + x3 }
(b) {x2 , x3 , 1 − 3x + 10x2 − 5x3 }
(c) {1, x2 − 2, x2 − 6}
(d) {x − 3x2 , x3 , x2 − 3x + 2, x − 1, 5x2 − 7}
(e) {1, x, x3 , x − x2 , 3x − 1, 2x3 }.
3. (10 pts.) Let B = {1, ex , e2x }. Show that B is LI and find the
coordinates of f (x) = (1 − 3ex )2 .
4. (15 pts.) Find bases for the column space, null space, and row space
of C, and state the rank and nullity of C. What should these sum to?
Do they?


1 −2 −1 −3
2
4 
C =  −1 2
2 −4 −3 −7
5. Let L : P2 → P2 be defined by L[p] = (x2 − 1)p00 + (x + 3)p0 − 4p .
(a) (5 pts.) Show that L is linear.
(b) (5 pts.) Find the matrix A of L relative to the basis B =
{1, x, x2 }.
(c) (5 pts.) Find a basis for the null space of L. What is the rank
of L?
6. (15 pts.) Consider the matrix A below. Find the eigenvalues and
eigenvectors of A. Also, find a diagonal matrix Λ and an invertible
matrix S for which A = SΛS −1 .


2 0 1
A= 0 3 0 .
1 0 2
Rπ
7. (15 pts.) Consider the inner product hf, gi = 0 f (x)g(x)dx on
continuous functions C[0, π]. Use the Gram-Schmidt process to turn
2
{1, cos(x),
(x)} into an orthogonal set relative
this inner product.
R ton−2
R cos
n−1
1
n−1
n
(Hint: cos (x)dx = n cos (x) sin(x) + n
cos (x)dx.)
R1
8. Consider the inner product hf, gi = −1 f (x)g(x)dx on continuous functions C[−1, 1].
(a) (5 pts.) Find the Gram matrix A for {1, x, x2 }.
(b) (10 pts.) Use A and the given inner product to find the best
quadratic fit to f (x) = |x|.
2